Assume we have a generator $Q$ describing the transitions between four states $A,B,C,D$. $Q$ is a valid generator and all non-diagonal elements are strictly positive hence no absorbing states. I am interested in the transition probability of moving from $A$ to $B$ in say $T=1$ year. Normally this is simply the matrix element $[\exp(Q)]_{AB}$.
However I want to calculate the probability such that it excludes paths that reach $D$ during the year. That is the transition probability should exclude the possibility of reaching $D$ during the year. How can I calculate this probability in terms of generator elements $Q$? In case state $D$ is absorbing the probability should simplify to $[\exp(Q)]_{AB}$.
Any help on how to derive the desired probability is much appreciated!
Update on possible answer: If we simply set the last row in $Q$ to zero and call this new generator $Q^*$ is the answer then simply $e^{Q*}$?